Cover
List of Figures
List of Tables
Preface
Acknowledgments
Part I: Neural Networks for Serial Robot Arm Control
1. 1 Zeroing Neural Networks for Control
  1. 1.1 Introduction
  2. 1.2 Scheme Formulation and ZNN Solutions
  3. 1.3 Theoretical Analyses
  4. 1.4 Computer Simulations and Verifications
  5. 1.5 Summary
2. 2 Adaptive Dynamic Programming Neural Networks for Control
  1. 2.1 Introduction
  2. 2.2 Preliminaries on Variable Structure Control of the Sensor–Actuator System
  3. 2.3 Problem Formulation
  4. 2.4 Model‐Free Control of the Euler–Lagrange System
  5. 2.5 Simulation Experiment
  6. 2.6 Summary
3. 3 Projection Neural Networks for Robot Arm Control
  1. 3.1 Introduction
  2. 3.2 Problem Formulation
  3. 3.3 A Modified Controller without Error Accumulation
  4. 3.4 Performance Improvement Using Velocity Compensation
  5. 3.5 Simulations
  6. 3.6 Summary
4. 4 Neural Learning and Control Co‐Design for Robot Arm Control
  1. 4.1 Introduction
  2. 4.2 Problem Formulation
  3. 4.3 Nominal Neural Controller Design
  4. 4.4 A Novel Dual Neural Network Model
  5. 4.5 Simulations
  6. 4.6 Summary
5. 5 Robust Neural Controller Design for Robot Arm Control
  1. 5.1 Introduction
  2. 5.2 Problem Formulation
  3. 5.3 Dual Neural Networks for the Nominal System
  4. 5.4 Neural Design in the Presence of Noises
  5. 5.5 Simulations
  6. 5.6 Summary
6. 6 Using Neural Networks to Avoid Robot Singularity
  1. 6.1 Introduction
  2. 6.2 Preliminaries
  3. 6.3 Problem Formulation
  4. 6.4 Reformulation as a Constrained Quadratic Program
  5. 6.5 Neural Networks for Redundancy Resolution
  6. 6.6 Illustrative Examples
  7. 6.7 Summary
Part II: Neural Networks for Parallel Robot Control
1. 7 Neural Network Based Stewart Platform Control
  1. 7.1 Introduction
  2. 7.2 Preliminaries
  3. 7.3 Robot Kinematics
  4. 7.4 Problem Formulation as Constrained Optimization
  5. 7.5 Dynamic Neural Network Model
  6. 7.6 Theoretical Results
  7. 7.7 Numerical Investigation
  8. 7.8 Summary
2. 8 Neural Network Based Learning and Control Co‐Design for Stewart Platform Control
  1. 8.1 Introduction
  2. 8.2 Kinematic Modeling of Stewart Platforms
  3. 8.3 Recurrent Neural Network Design
  4. 8.4 Numerical Investigation
  5. 8.5 Summary
Part III: Neural Networks for Cooperative Control
1. 9 Zeroing Neural Networks for Robot Arm Motion Generation
  1. 9.1 Introduction
  2. 9.2 Preliminaries
  3. 9.3 Problem Formulation and Distributed Scheme
  4. 9.4 NTZNN Solver and Theoretical Analyses
  5. 9.5 Illustrative Examples
  6. 9.6 Summary
2. 10 Zeroing Neural Networks for Robot Arm Motion Generation
  1. 10.1 Introduction
  2. 10.2 Preliminaries, Problem Formulation, and Distributed Scheme
  3. 10.3 NANTZNN Solver and Theoretical Analyses
  4. 10.4 Illustrative Examples
  5. 10.5 Summary
References
Index
End User License Agreement

List of Tables

Chapter 1
1. Table 1.1 Comparison of ZNN‐based and gradient‐based techniques for solving
Chapter 3
1. Table 3.1 Summary of the Denavit–Hartenberg parameters of the PUMA 560 manipu...
2. Table 3.2 Comparisons of different RNN algorithms for the tracking control of...
3. Table 3.3 The RMS position tracking errors of the different controllers.
Chapter 4
1. Table 4.1 Summary of the Denavit–Hartenberg parameters of the PUMA 560 manipu...
Chapter 5
1. Table 5.1 Summary of the Denavit–Hartenberg parameters of the PUMA manipula...
Chapter 7
1. Table 7.1 Comparisons of different methods for kinematic control of Stewart p...
Chapter 9
1. Table 9.1 Comparison of different schemes for redundancy resolution of manipu...

List of Illustrations

Chapter 1
1. Figure 1.1 State vector of ZNN model (1.8) for solving (1.13) at . (a) Pro...
2. Figure 1.2 State vector of ZNN model (1.8) for solving (1.13) at . (a) Pro...
3. Figure 1.3 Residual error of ZNN model (1.8) for solving (1.13) at .
4. Figure 1.4 State vector of ZNN model (1.8) for solving (1.13) at . (a) Pro...
5. Figure 1.6 Residual error of ZNN model (1.8) for solving (1.13) at .
6. Figure 1.7 State vector of ZNN model (1.12) for solving (1.16). (a) Profile...
7. Figure 1.8 State vector of ZNN model (1.12) for solving (1.16). (a) Profile...
8. Figure 1.9 Residual error of ZNN model (1.12) for solving (1.16).
Chapter 2
1. Figure 2.1 The cart–pole system.
2. Figure 2.2 State profiles of the cart–pole system with the proposed control s...
Chapter 3
1. Figure 3.1 Simulation results for the position regulation control of the end‐...
2. Figure 3.2 Simulation results for the position regulation control of the end‐...
3. Figure 3.3 Simulation results for the tracking control of the end‐effector of...
4. Figure 3.4 Simulation results for the tracking control of the end‐effector of...
5. Figure 3.5 Simulation results obtained using the presented control laws with ...
6. Figure 3.6 Simulation results obtained using the presented control laws with ...
7. Figure 3.7 Simulation comparisons for the tracking control of the end‐effecto...
8. Figure 3.12 Simulation comparisons for the tracking control of the end‐effect...
Chapter 4
1. Figure 4.1 The neural connections between different neural states in the prop...
2. Figure 4.2 The control block diagram of the overall system using the proposed...
3. Figure 4.3 A schematic of the PUMA 560 manipulator.
4. Figure 4.4 The trajectory of the manipulator end‐effector using the proposed ...
5. Figure 4.5 Simulation results using the proposed algorithm. Time history of (...
6. Figure 4.6 Simulation results using the proposed algorithm. Time history of (...
7. Figure 4.7 Simulation results using the proposed algorithm. Time history of (...
8. Figure 4.8 Simulation results using the proposed algorithm. Time history of (...
9. Figure 4.9 The trajectory of the manipulator end‐effector using the algorithm...
10. Figure 4.10 The trajectory of the manipulator end‐effector using the algorith...
11. Figure 4.11 The position tracking trajectory of the manipulator end‐effector ...
12. Figure 4.12 The position tracking trajectory of the manipulator end‐effector ...
Chapter 5
1. Figure 5.1 The architecture of the presented recurrent neural networks.
2. Figure 5.2 The control diagram of the system using the presented neural contr...
3. Figure 5.3 The schematic of a 6‐DOF PUMA 560 manipulator considered in the si...
4. Figure 5.4 The time profile of control parameters for the presented neural ne...
5. Figure 5.5 Comparisons of the presented approach with existing ones in the pr...
6. Figure 5.6 Comparisons of the presented approach with existing ones in the pr...
7. Figure 5.7 Comparisons of the presented approach with existing ones in the pr...
8. Figure 5.8 Performance comparisons of the presented approach for the tracking...
Chapter 6
1. Figure 6.1 Neural network architecture.
2. Figure 6.2 Simulation results on (a) motion trajectories and (b) manipulabili...
3. Figure 6.3 Simulation results of (a) joint‐angle, (b) joint‐velocity and (c) ...
4. Figure 6.4 Simulation results of (a) motion trajectories, (b) manipulability ...
5. Figure 6.5 Simulation results of PUMA 560 for tracking a circular path in the...
6. Figure 6.6 Manipulability measures of PUMA 560 by different schemes.
Chapter 7
1. Figure 7.1 Tracking of a circular motion. (a) The trajectory of the end‐effec...
2. Figure 7.2 The time evolution of the Stewart platform state variables in the ...
3. Figure 7.3 Simulation results on motion trajectories, manipulability measures...
4. Figure 7.4 The time evolution of the Stewart platform state variables in the ...
5. Figure 7.5 The time evolution of the Stewart platform state variables in the ...
6. Figure 7.6 The time evolution of the neural network state variables in the ca...
7. Figure 7.7 Tracking of a square motion. (a) End‐effector trajectory and (b) p...
8. Figure 7.8 The time evolution of the Stewart platform state variables in the ...
9. Figure 7.9 The time evolution of the neural network state variables in the ca...
Chapter 8
1. Figure 8.1 Schematic of a Stewart platform.
2. Figure 8.2 Stewart platform geometric representation. The gray triangle at th...
3. Figure 8.3 Architecture of the proposed neural network.
4. Figure 8.4 Tracking of a circular motion: (a) The tracking trajectory of the ...
5. Figure 8.5 The time evolution of the Stewart platform state variables in the ...
6. Figure 8.6 Tracking of a square motion. (a) End‐effector trajectory and (b) p...
7. Figure 8.7 The time evolution of the platform state variables in the case of ...
Chapter 9
1. Figure 9.1 Circuit schematic of the distributed NTZNN model (9.13).
2. Figure 9.2 Control diagram of the MVN‐oriented distributed scheme (9.8) aided...
3. Figure 9.3 Computer simulations synthesized by MVN‐oriented distributed schem...
4. Figure 9.4 Three‐dimensional view of motion trajectories of eight PUMA 560 ro...
5. Figure 9.5 Computer simulations synthesized by MVN‐oriented distributed schem...
6. Figure 9.6 Computer simulations synthesized by MVN‐oriented distributed schem...
Chapter 10
1. Figure 10.1 Computer simulations synthesized by MVN‐oriented distributed sche...
2. Figure 10.2 Computer simulations synthesized by MVN‐oriented distributed sche...
3. Figure 10.3 Computer simulations synthesized by MVN‐oriented distributed sche...
4. Figure 10.4 Computer simulations synthesized by MVN‐oriented distributed sche...
5. Figure 10.5 Computer simulations synthesized by MVN‐oriented distributed sche...

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Library of Congress Cataloging-in-Publication Data

Names: Li, Shuai, 1983– author. | Jin, Long, 1988– author. | Mirza, Mohammed Aquil, 1986– author.
Title: Kinematic control of redundant robot arms using neural networks / Shuai Li, Hong Kong Polytechnic University, Long Jin, Hong Kong Polytechnic University, Mohammed Aquil Mirza, Hong Kong Polytechnic University.
Description: First edition. | Hoboken, NJ : John Wiley & Sons, Inc., 2019. | “Most of the materials of this book are derived from the authors’ papers published in journals and proceedings of the international conferences”–Introduction. | Includes bibliographical references and index. |
Identifiers: LCCN 2018048730 (print) | LCCN 2018050945 (ebook) | ISBN 9781119556985 (Adobe PDF) | ISBN 9781119556992 (ePub) | ISBN 9781119556961 (hardcover)
Subjects: LCSH: Robots–Kinematics–Data processing. | Manipulators (Mechanism)–Automatic control. | Redundancy (Engineering)–Data processing. | Neural networks (Computer science)
Classification: LCC TJ211.412 (ebook) | LCC TJ211.412 .K563 2019 (print) | DDC 629.8/95632–dc23
LC record available at https://lccn.loc.gov/2018048730

List of Figures

1.1	State vector of ZNN model (1.8) for solving (1.13) at . (a) Profiles of and (b) profile of .
1.2	State vector of ZNN model (1.8) for solving (1.13) at . (a) Profiles of and (b) profile of .
1.3	Residual error of ZNN model (1.8) for solving (1.13) at .
1.4	State vector of ZNN model (1.8) for solving (1.13) at . (a) Profiles of and (b) profile of .
1.5	State vector of ZNN model (1.8) for solving (1.13) at . (a) Profiles of and (b) profiles of .
1.6	Residual error of ZNN model (1.8) for solving (1.13) at .
1.7	State vector of ZNN model (1.12) for solving (1.16). (a) Profiles of and (b) profile of .
1.8	State vector of ZNN model (1.12) for solving (1.16). (a) Profiles of and (b) profiles of .
1.9	Residual error of ZNN model (1.12) for solving (1.16).
2.1	The cart–pole system.
2.2	State profiles of the cart–pole system with the proposed control strategy.
3.1	Simulation results for the position regulation control of the end‐effector of a PUMA 560 to maintain a fixed position of m in the workspace. (a) The end‐effector trajectory and (b) the time history of the joint angle .
3.2	Simulation results for the position regulation control of the end‐effector of a PUMA 560 to maintain a fixed position of m in the workspace. Time history of (a) the control error and (b) the control action .
3.3	Simulation results for the tracking control of the end‐effector of a PUMA 560 with respect to a time‐varying reference along a circular path. (a) The end‐effector trajectory and (b) the time history of the joint angle .
3.4	Simulation results for the tracking control of the end‐effector of a PUMA 560 with respect to a time‐varying reference along a circular path. Time history of (a) the control error and (b) the control action .
3.5	Simulation results obtained using the presented control laws with a nonconvex projection set. Time history of (a) the control error with (3.19) and (b) the control action with (3.19).
3.6	Simulation results obtained using the presented control laws with a nonconvex projection set. Time history of (a) the control error with (3.41) and (b) the control action with (3.41).
3.7	Simulation comparisons for the tracking control of the end‐effector of a PUMA 560 with respect to a time‐varying reference along a circular path using different neuro‐controllers, i.e. controller 1 (control law (3.19) presented in this chapter), controller 2 (control law (3.41) presented in this chapter), controller 3 [40], and controller 4 [39], in the presence of random Gaussian noise at different levels of . (a) Controller 1, noise level and (b) controller 1, noise level
3.8	Simulation comparisons for the tracking control of the end‐effector of a PUMA 560 with respect to a time‐varying reference along a circular path using different neuro‐controllers, i.e. controller 1 (control law (3.19) proposed in this chapter), controller 2 (control law (3.41) presented in this chapter), controller 3 [40], and controller 4 [39], in the presence of random Gaussian noise at different levels of . (a) Controller 1, noise level and (b) controller 2, noise level .
3.9	Simulation comparisons for the tracking control of the end‐effector of a PUMA 560 with respect to a time‐varying reference along a circular path using different neuro‐controllers, i.e. controller 1 (control law (3.19) proposed in this chapter), controller 2 (control law (3.41) presented in this chapter), controller 3 [40], and controller 4 [39], in the presence of random Gaussian noise at different levels of . (a) Controller 2, noise level and (b) controller 2, noise level .
3.10	Simulation comparisons for the tracking control of the end‐effector of a PUMA 560 with respect to a time‐varying reference along a circular path using different neuro‐controllers, i.e. controller 1 (control law (3.19) proposed in this chapter), controller 2 (control law (3.41) presented in this chapter), controller 3 [40], and controller 4 [39], in the presence of random Gaussian noise at different levels of . (a) Controller 3, noise level and (b) controller 3, noise level .
3.11	Simulation comparisons for the tracking control of the end‐effector of a PUMA 560 with respect to a time‐varying reference along a circular path using different neuro‐controllers, i.e. controller 1 (control law (3.19) proposed in this chapter), controller 2 (control law (3.41) presented in this chapter), controller 3 [40], and controller 4 [39], in the presence of random Gaussian noise at different levels of . (a) Controller 3, noise level and (b) controller 4, noise level .
3.12	Simulation comparisons for the tracking control of the end‐effector of a PUMA 560 with respect to a time‐varying reference along a circular path using different neuro‐controllers, i.e. controller 1 (control law (3.19) presented in this chapter), controller 2 (control law (3.41) presented in this chapter), controller 3 [40], and controller 4 [39], in the presence of random Gaussian noise at different levels of . (a) Controller 4, noise level and (b) controller 4, noise level .
4.1	The neural connections between different neural states in the proposed neural controller.
4.2	The control block diagram of the overall system using the proposed neural network for the control of a manipulator.
4.3	A schematic of the PUMA 560 manipulator.
4.4	The trajectory of the manipulator end‐effector using the proposed algorithm with excitation noises, where the piecewise straight lines represent the links of the manipulator and the curve represents the trajectory of the end‐effector. (a) – view; (b) – view; and (c) – view.
4.5	Simulation results using the proposed algorithm. Time history of (a) and () .
4.6	Simulation results using the proposed algorithm. Time history of (a) all elements of the estimated Jacobian matrix and (b) the Jacobian estimation error .
4.7	Simulation results using the proposed algorithm. Time history of (a) the position error and (b) the resolved velocity error .
4.8	Simulation results using the proposed algorithm. Time history of (a) the co‐state and (b) .
4.9	The trajectory of the manipulator end‐effector using the algorithm without additive noises, where the piecewise straight lines represent the links of the manipulator and the curve represents the trajectory of the end‐effector. (a) – view; (b) – view; and (c) – view.
4.10	The trajectory of the manipulator end‐effector using the algorithm without additive noises. Time history of (a) the estimation error for the Jacobian matrix; (b) the position tracking error; and (c) the velocity tracking error.
4.11	The position tracking trajectory of the manipulator end‐effector under different levels of measurement noises of . (a) and (b) .
4.12	The position tracking trajectory of the manipulator end‐effector under different levels of measurement noises of . (a) and (b) .
5.1	The architecture of the presented recurrent neural networks.
5.2	The control diagram of the system using the presented neural controller to steer the motion of a redundant manipulator.
5.3	The schematic of a 6‐DOF PUMA 560 manipulator considered in the simulation and the generated trajectory of PUMA 560 controlled by the presented neural network in the absence of noises. (a) PUMA 560 manipulator and (b) end‐effector trajectory.
5.4	The time profile of control parameters for the presented neural network in the absence of noises. (a) Position tracking error and (b) joint angles.
5.5	Comparisons of the presented approach with existing ones in the presence of different levels of noises with noise . (a) This chapter; (b) [60]; and (c) [39].
5.6	Comparisons of the presented approach with existing ones in the presence of different levels of noises with noise . (a) This chapter; (b) [60]; and (c) [39].
5.7	Comparisons of the presented approach with existing ones in the presence of different levels of noises with noise . (a) This chapter; (b) [60]; and (c) [39].
5.8	Performance comparisons of the presented approach for the tracking of circular motions in the presence of various noises. (a) Linear noise ; (b)quadratic noise ; (c) cubic noise ; (d) fourth‐order noise ; and (e) noise .
6.1	Neural network architecture.
6.2	Simulation results on (a) motion trajectories and (b) manipulability measures for the manipulability optimization of PUMA 560 manipulator via self motion with its end‐effector fixed at [0.55, 0, 1.3] m in the workspace.
6.3	Simulation results of (a) joint‐angle, (b) joint‐velocity and (c) position‐error profiles for the manipulability optimization of PUMA 560 manipulator via self motion with its end‐effector fixed at [0.55, 0, 1.3] m in the workspace.
6.4	Simulation results of (a) motion trajectories, (b) manipulability measures and (c) joint‐angle profiles of PUMA 560 synthesized by the scheme presented in [72] with its end‐effector fixed at [0.55, 0, 1.3] m in the workspace.
6.5	Simulation results of PUMA 560 for tracking a circular path in the workspace synthesized by the proposed scheme (6.32). (a) Motion trajectories; (b) manipulability measures; (c) joint‐angle profiles; and (d) joint‐velocity profiles.
6.6	Manipulability measures of PUMA 560 by different schemes.
7.1	Tracking of a circular motion. (a) The trajectory of the end‐effector and (b) the time history of the position tracking error.
7.2	The time evolution of the Stewart platform state variables in the case of circular motion tracking. (a) Orientation of the platform ; (b) position of the end‐effector ; (c) control action ; and (d) leg length .
7.3	Simulation results on motion trajectories, manipulability measures and joint‐angle profiles of PUMA 560 synthesized by the scheme presented in [71] with its end‐effector fixed at [0.55, 0, 1.3] m in the workspace. (a) State variable ; (b) state variable ; and (c) state variable .
7.4	The time evolution of the Stewart platform state variables in the case of circular motion tracking. (a) End‐effector trajectory and (b) position tracking error.
7.5	The time evolution of the Stewart platform state variables in the case of infinity‐sign motion tracking. (a) Orientation of the platform ; (b) position of the end‐effector ; (c) control action ; and (d) leg length .
7.6	The time evolution of the neural network state variables in the case of infinity‐sign motion tracking. (a) State variable ; (b) state variable ; and (c) state variable .
7.7	Tracking of a square motion. (a) End‐effector trajectory and (b) position tracking error.
7.8	The time evolution of the Stewart platform state variables in the case of infinity‐sign motion tracking. (a) Orientation of the platform ; (b) position of the end‐effector ; (c) control action ; and (d) leg length .
7.9	The time evolution of the neural network state variables in the case of square motion tracking. (a) State variable ; (b) state variable ; and (c) state variable .
8.1	Schematic of a Stewart platform.
8.2	Stewart platform geometric representation. The gray triangle at the top and the gray hexagon at the bottom represent the moving top plate and the fixed base plate, respectively.
8.3	Architecture of the proposed neural network.
8.4	Tracking of a circular motion: (a) The tracking trajectory of the end‐effector and (b) the time history of the position tracking error.
8.5	The time evolution of the Stewart platform state variables in the case of circular motion tracking. (a) Orientation of the platform ; (b) control action ; and (c) leg length .
8.6	Tracking of a square motion. (a) End‐effector trajectory and (b) position tracking error.
8.7	The time evolution of the platform state variables in the case of square motion tracking. (a) Orientation of the platform ; (b) control action ; and (c) leg length .
9.1	Circuit schematic of the distributed NTZNN model (9.13).
9.2	Control diagram of the MVN‐oriented distributed scheme (9.8) aided by the distributed NTZNN model (9.13) for the cooperative motion generation of a network of redundant robot manipulators.
9.3	Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) and distributed NTZNN model (9.13) for consensus of eight redundant PUMA 560 robot manipulators with limited communications and perturbed with constant noise , where initial joint states of manipulators are randomly generated and R denotes the ith redundant robot manipulator. (a) Motion trajectories; (b) profiles of end‐effector position errors; (c) profiles of joint angle; and (d) profiles of joint velocity.
9.4	Three‐dimensional view of motion trajectories of eight PUMA 560 robot manipulators synthesized by MVN‐oriented distributed scheme (9.8) as well as distributed NTZNN model (9.13) perturbed with noise for cooperative payload transport along a time‐varying circular reference.
9.5	Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) as well as distributed NTZNN model (9.13) perturbed with noise for cooperative payload transport of eight PUMA 560 robot manipulators along a time‐varying circular reference with limited communications. Profiles of (a) position errors, (b) joint angle, and (c) joint velocity.
9.6	Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) as well as distributed ZNN model perturbed with noise for cooperative payload transport of eight PUMA 560 robot manipulators along a time‐varying circular reference with limited communications. Profiles of (a) end‐effector position errors, (b) joint angle, and (c) joint velocity.
10.1	Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) and distributed NTZNN model (9.13) for consensus of eight redundant PUMA 560 robot manipulators with limited communications and perturbed with constant noise , where initial joint states of manipulators are randomly generated. (a) Motion trajectories; (b) profiles of end‐effector position errors; (c) profiles of joint angle; and (d) profiles of .
10.2	Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) and distributed NTZNN model (9.13) for consensus of eight redundant PUMA 560 robot manipulators with limited communications and perturbed with constant noise , where initial joint states of manipulators are randomly generated. Profiles of (a) joint angle and (b) joint velocity.
10.3	Computer simulations synthesized by MVN‐oriented distributed scheme (10.8) and hyperbolic‐sine activation function activated NANTZNN model (10.12) perturbed with noise for motion planning of 6 redundant PUMA 560 robot arms with limited communications. (a) Motion trajectories; (b) profiles of end‐effector position errors; (c) profiles of joint angle; and (d) profiles of .
10.4	Computer simulations synthesized by MVN‐oriented distributed scheme (10.8) and hyperbolic‐sine activation function activated NANTZNN model (10.12) perturbed with noise for motion planning of six redundant PUMA 560 robot arms with limited communications. Profiles of (a) joint angle and (b) joint velocity.
10.5	Computer simulations synthesized by MVN‐oriented distributed scheme (10.8) and power‐sigmoid activation function activated NANTZNN model (10.12) perturbed with noise for motion planning of six redundant PUMA 560 robot arms with limited communications. Profiles of (a) end‐effector position errors and (b) joint angle.

1.1	Comparison of ZNN‐based and gradient‐based techniques for solving .
3.1	Summary of the Denavit–Hartenberg parameters of the PUMA 560 manipulator used in the simulation.
3.2	Comparisons of different RNN algorithms for the tracking control of a PUMA 560 manipulator.
3.3	The RMS position tracking errors of the different controllers.
4.1	Summary of the Denavit–Hartenberg parameters of the PUMA 560 manipulator used in the simulation.
5.1	Summary of the Denavit–Hartenberg parameters of the PUMA manipulator used in the simulation.
7.1	Comparisons of different methods for kinematic control of Stewart platforms.
9.1	Comparison of different schemes for redundancy resolution of manipulators.